Column - I Column - IIA)Radius of the largest circle which passes through The focus of the parabola y2 = 4x and contained in it, isP)16B)Two perpendicular tangents PA and PB are drawn to the parabola y2 =16x then length AB may beQ)5C)The shortest distance between parabolas y2 = 4x and y2 = 2x - 6 is d then d2R)8D)The harmonic mean of the segments of a focal chord of Parabola y2 = 8xS)4 ABCD1.PQSS2.SPQS3.PSQS4.SQPS

x2+y2+2gx+c=0,y2=4x,Passing through S (1, 0) ⇒c = -1-2g⇒x2+(2g+4)x−2g−1=0Δ=0⇒g=−1( or −5), If g=−5⇒c=9∴r=25−9=16=4 B) AB=L⋅L⋅R=4a=16 C) y2=4x (1), y2=2x−6 (2) Let ‘p’ be any point on (1) ∴P=t2,2tLet ‘Q’ be any point on (2) ∴Q=3+t22,tPQ2=f(t)=t2−624+t2‘f’ is minimum if f(t) = 0⇒t2 = 4⇒d2=(PQ)2=5 D) 2SP⋅SQSP+SQ=SL=4

Column - I Column - IIA)Radius of the largest circle which passes through The focus of the parabola y2 = 4x and contained in it, isP)16B)Two perpendicular tangents PA and PB are drawn to the parabola y2 =16x then length AB may beQ)5C)The shortest distance between parabolas y2 = 4x and y2 = 2x - 6 is d then d2R)8D)The harmonic mean of the segments of a focal chord of Parabola y2 = 8xS)4 ABCD1.PQSS2.SPQS3.PSQS4.SQPS

x2+y2+2gx+c=0,y2=4x,Passing through S (1, 0) ⇒c = -1-2g⇒x2+(2g+4)x−2g−1=0Δ=0⇒g=−1( or −5), If g=−5⇒c=9∴r=25−9=16=4 B) AB=L⋅L⋅R=4a=16 C) y2=4x (1), y2=2x−6 (2) Let ‘p’ be any point on (1) ∴P=t2,2tLet ‘Q’ be any point on (2) ∴Q=3+t22,tPQ2=f(t)=t2−624+t2‘f’ is minimum if f(t) = 0⇒t2 = 4⇒d2=(PQ)2=5 D) 2SP⋅SQSP+SQ=SL=4

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Column - I Column - II
A) Radius of the largest circle which passes through The focus of the parabola y² = 4x and contained in it, is P) 16
B) Two perpendicular tangents PA and PB are drawn to the parabola y² =16x then length AB may be Q) 5
C) The shortest distance between parabolas y² = 4x and y² = 2x - 6 is d then d² R) 8
D) The harmonic mean of the segments of a focal chord of Parabola y² = 8x S) 4
A B C D
1. P Q S S
2. S P Q S
3. P S Q S
4. S Q P S

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Detailed Solution

$x^{2} + y^{2} + 2 gx + c = 0, y^{2} = 4 x,$
Passing through S (1, 0) $\Rightarrow$ c = -1-2g
$\begin{array}{l} \Rightarrow x^{2} + (2 g + 4) x - 2 g - 1 = 0 \\ Δ = 0 \Rightarrow g = - 1 (or - 5), If g = - 5 \Rightarrow c = 9 \\ ∴ r = \sqrt{25 - 9} = \sqrt{16} = 4 \\ B) AB = L \cdot L \cdot R = 4 a = 16 \\ C) y^{2} = 4 x (1), y^{2} = 2 x - 6 (2) \end{array}$
Let ‘p’ be any point on (1) $∴ P = (t^{2}, 2 t)$
Let ‘Q’ be any point on (2) $∴ Q = (3 + \frac{t^{2}}{2}, t)$
${PQ}^{2} = f (t) = \frac{{(t^{2} - 6)}^{2}}{4} + t^{2}$
‘f’ is minimum if f(t) = 0 $\Rightarrow$ t² = 4
$\Rightarrow d^{2} = (PQ)^{2} = 5 D) \frac{2 SP \cdot SQ}{SP + SQ} = SL = 4$

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	Column - I		Column - II
A)	Radius of the largest circle which passes through The focus of the parabola y² = 4x and contained in it, is	P)	16
B)	Two perpendicular tangents PA and PB are drawn to the parabola y² =16x then length AB may be	Q)	5
C)	The shortest distance between parabolas y² = 4x and y² = 2x - 6 is d then d²	R)	8
D)	The harmonic mean of the segments of a focal chord of Parabola y² = 8x	S)	4